1. Principles of Plane Wave Interferometric Sensor
1.1. Principles of Classical Plane Wave Fabry-Perot Interferometric Sensor
The theory of Fabry-Perot sensor is based on classical Fabry-Perot interferometry of plane waves (FIG. 1). Due to interference of multiply reflected beams, the intensity of the total reflected light I(o) is expressed as Airy function
                              I                      (            o            )                          =                                            F              ⁢                                                          ⁢                              sin                2                            ⁢                              δ                2                                                    1              +                              F                ⁢                                                                  ⁢                                  sin                  2                                ⁢                                  δ                  2                                                              ⁢                      I                          (              i              )                                                          (        1        )            where I(i) is the intensity of the incident light, and δ the phase dependent on the optic path or interference gap width L. F is the finesse, defined by
                    F        =                              4            ⁢            R                                              (                              1                -                R                            )                        2                                              (        2        )            where
                    R        =                                                                        r                ′                            ⁢                              r                ″                                                                                    (        3        )            r′ and r″ being the reflection coefficient at the interface of the media with n and n′, and that with n and n″, respectively. When F is small, say F<0.2, which corresponds to R<0.046, equation (1) can be approximated as [1]
                                                        I                              (                o                )                                                    I                              (                i                )                                              ≈                      F            ⁢                                                  ⁢                          sin              2                        ⁢                          δ              2                                      =                                            F              2                        ⁢                          (                              1                -                                  cos                  ⁢                                                                          ⁢                  δ                                            )                                =                                    F              2                        ⁡                          [                              1                +                                  sin                  ⁡                                      (                                                                                                                        4                            ⁢                            π                            ⁢                                                                                                                  ⁢                            n                                                    λ                                                ⁢                        L                                            +                                              ϕ                        o                                                              )                                                              ]                                                          (        4        )            where λ is the nominal wavelength of the light that generates the optic path differences of the series of reflected beams 2, n the refractive index of the medium of the gap (n˜1 for air, 1.33 for water, and 1.48 for oil), L the width of interference gap, and φo a phase factor related to the equilibrium gap width without input signal. Note that (4) depicts I(o) as a harmonic function of L, based on which Fabry-Perot interferometric sensor is designed.1.2. Principles of Classical Plane Wave Michelson/Mach-Zehnder Interferometric Sensor
The principles of Michelson interferometric sensor can be depicted by FIG. 2. FIG. 2 shows source 202, beamsplitter 204, mirrors 206 and detector 208. Instead of interference of multiply reflected beams of the Fabry-Perot interferometer, the output light intensity of the Michelson interferometer is due to interference of two beams [2]. One beam contains a variable optic path under measurement, and the other is the reference. The output light intensity I(o) of an ideal lossless Michelson interferometer is expressed as
                              I                      (            o            )                          =                              I            1                          (              i              )                                +                      I            2                          (              i              )                                +                      2            ⁢                                                            I                  1                                      (                    i                    )                                                  ⁢                                  I                  2                                      (                    i                    )                                                                        ⁢                          sin              ⁡                              (                                                                                                    4                        ⁢                        π                        ⁢                                                                                                  ⁢                        n                                            λ                                        ⁢                    L                                    +                                      ϕ                    o                                                  )                                                                        (        5        )            where I1(i) and I2(i) are the intensity of the probing beam and the reference beam, respectively. When I1(i)=I2(i)=I(i), (5) is reduced to
                              I                      (            o            )                          =                              2            ⁢                          I                              (                i                )                                              +                      2            ⁢                          I                              (                i                )                                      ⁢                          sin              ⁡                              (                                                                                                    4                        ⁢                        π                        ⁢                                                                                                  ⁢                        n                                            λ                                        ⁢                    L                                    +                                      ϕ                    o                                                  )                                                                        (        6        )            
A third type of optical interference device is Mach-Zehnder interferometric sensor (FIG. 3). In a Michelson type interferometric device there is only one beam splitter or coupler, while both the sensing leg and the reference leg reflect back from the mirrors. On the other hand, in a Mach-Zehnder type device there are two beam splitters or couplers, while neither the sensing leg nor the reference leg reflect back from the mirrors. Ideally they both observe equation (6). Therefore, these are referred to herein as Michelson/Mach-Zehnder interferometric sensors.
1.3. Comparison of Plane Wave Fabry-Perot and Michelson/Mach-Zehnder Interferometric Sensors
As shown in FIGS. 1, 2 and 3, 3 dB or 50%-50% beam splitters are necessary to construct the Fabry-Perot or Michelson/Mach-Zehnder interferometric sensors. FIG. 3 includes source 302, beamsplitters 304, mirrors 306, and detector 310. The meaningful way is to define the light source output intensity as the input intensity I(in) of the interferometric sensor, and the light intensity received by the detector as the output intensity I(out). Thus, an ideal Fabry-Perot interferometric sensor has
                              I                      (            out            )                          ≈                                            F              8                        ⁡                          [                              1                +                                  sin                  ⁡                                      (                                                                                                                        4                            ⁢                            π                            ⁢                                                                                                                  ⁢                            n                                                    λ                                                ⁢                        L                                            +                                              ϕ                        o                                                              )                                                              ]                                ⁢                      I                          (                              i                ⁢                                                                  ⁢                n                            )                                      <                                            1              40                        ⁢                          I                              (                                  i                  ⁢                                                                          ⁢                  n                                )                                              +                                    1              40                        ⁢                          sin              ⁡                              (                                                                                                    4                        ⁢                        π                        ⁢                                                                                                  ⁢                        n                                            λ                                        ⁢                    L                                    +                                      ϕ                    o                                                  )                                      ⁢                          I                              (                                  i                  ⁢                                                                          ⁢                  n                                )                                                                        (        7        )            while an ideal Michelson/Mach-Zehnder interferometric sensor has
                              I                      (            out            )                          =                                            1              2                        ⁡                          [                              1                +                                  sin                  ⁡                                      (                                                                                                                        4                            ⁢                            π                            ⁢                                                                                                                  ⁢                            n                                                    λ                                                ⁢                        L                                            +                                              ϕ                        o                                                              )                                                              ]                                <                                                    1                2                            ⁢                              I                                  (                                      i                    ⁢                                                                                  ⁢                    n                                    )                                                      +                                          1                2                            ⁢                              sin                ⁡                                  (                                                                                                              4                          ⁢                          π                          ⁢                                                                                                          ⁢                          n                                                λ                                            ⁢                      L                                        +                                          ϕ                      o                                                        )                                            ⁢                              I                                  (                                      i                    ⁢                                                                                  ⁢                    n                                    )                                                                                        (        8        )            The efficiency in terms of I(out)/I(in) of a Michelson/Mach-Zehnder type interferometric sensor is more than 20 times higher than that of a Fabry-Perot type interferometric sensor.